A box from the annus mirabilis —
See Hudson's 4×4 array.
Related material —
See also "Romancing the Omega" —
Related mathematics — Guitart in this journal —
See also Weyl + Palermo in this journal —
The previous post suggests two images …
The date of the above Promoted Tweet is October 19, 2015.
"So we beat on, boats against the current …" — F. Scott Fitzgerald
The previous post, on subjective and objective quality,
suggests a review of Pirsig —
"And finally: Phaedrus, following a path
that to his knowledge had never been taken before
in the history of Western thought,
went straight between the horns of
the subjectivity-objectivity dilemma and said
Quality is neither a part of mind, nor is it a part of matter.
It is a third entity which is independent of the two.
He was heard along the corridors
and up and down the stairs of Montana Hall
singing softly to himself, almost under his breath,
'Holy, holy, holy…blessed Trinity.' "
See The Diamond Archetype and a fictional account of the road to Hell …
The cover illustration below has been adapted to
replace the flames of PyrE with the eightfold cube.
"… the A B C of being…." — Wallace Stevens
Compare to my own later note, from March 4, 2010 —
Elijah Wood in "The Last Witch Hunter" —
See Graham Priest in this journal.
See also Coxeter + Discursive.
The New York Times has a readable, if not informative,
review of a recent controversial account of history —
"For many, it exists in a kind of liminal state,
floating somewhere between fact and mythology."
[See Wikipedia on Liminality.]
Mahler begins his review with a statement by the President
on the night of May 1, 2011.
A more easily checked statement quoted here on that date:
"The positional meaning of a symbol derives from
its relationship to other symbols in a totality, a Gestalt,
whose elements acquire their significance from the
system as a whole."
A Gestalt from "Verhexung ," the previous post —
Guitart's statement that the above figure is a "Boolean logical cube"
seems, in the words of the Times , to be "floating somewhere
between fact and mythology." Discuss.
"Die Philosophie ist ein Kampf gegen die Verhexung
unsres Verstandes durch die Mittel unserer Sprache."
— Philosophical Investigations (1953), Section 109
An example of Verhexung from the René Guitart article in the previous post —
See also Ein Kampf .
This post continues recent thoughts on the work of René Guitart.
A 2014 article by Guitart gives a great deal of detail on his
approach to symmetric generation of the simple group of order 168 —
"Hexagonal Logic of the Field F8 as a Boolean Logic
with Three Involutive Modalities," pp. 191-220 in
The Road to Universal Logic:
Festschrift for 50th Birthday of
Jean-Yves Béziau, Volume I,
See also the eightfold cube in this journal.
According to René Guitart in May 2008 —
"In fact, in concrete terms, the Mathematical Pulsation is
nothing else but the thing that everyone does when doing
mathematics, even the most elementary ones. It is a very
special gesture in understanding ('geste de pensée'), well
known by each mathematician. The mind have to go to
and fro between to antinomical postures: to have the
situation under control, to leave the door open. To master
and to fix (a clear unique meaning) or to neglect and to
change (toward other possible meanings). Because of the
similarity of the pulsation of inspiration and expiration in
breath with the pulsation of closing and opening phases
in mathematical thinking, at the end of [Guitart (2003/a)]
I suggested to consider the famous book 'Zen in the Art
of Archery' [Herrigel (1997)] as a true treatise in didactic
of mathematics: just you have to replace everywhere the
words 'archery' by 'mathematical proof'."
Update of 6:20 AM Oct. 19, 2015 —
„Ich begriff plötzlich, daß in der Sprache oder doch
mindestens im Geist des Glasperlenspiels tatsächlich
alles allbedeutend sei, daß jedes Symbol und jede
Kombination von Symbolen nicht hierhin oder dorthin,
nicht zu einzelnen Beispielen, Experimenten und
Beweisen führe, sondern ins Zentrum, ins Geheimnis
und Innerste der Welt, in das Urwissen. Jeder Übergang
von Dur zu Moll in einer Sonate, jede Wandlung eines
Mythos oder eines Kultes, jede klassische, künstlerische
Formulierung sei, so erkannte ich im Blitz jenes
Augenblicks, bei echter meditativer Betrachtung,
nichts andres als ein unmittelbarer Weg ins Innere
des Weltgeheimnisses, wo im Hin und Wider zwischen
Ein- und Ausatmen, zwischen Himmel und Erde,
zwischen Yin und Yang sich ewig das Heilige vollzieht.“
— Hermann Hesse, Das Glasperlenspiel.
Berlin: Suhrkamp Taschenbuch Verlag, 2012. p. 172,
as quoted in a weblog.
For a version in English, see Summa Mythologica (Nov. 3, 2009).
From slides dated June 28, 2008 —
Compare to my own later note, from March 4, 2010 —
It seems that Guitart discovered these "A, B, C" generators first,
though he did not display them in their natural setting,
the eightfold cube.
Some context: The epigraph to my webpage
"A Simple Reflection Group of Order 168" —
"Let G be a finite, primitive subgroup of GL(V) = GL(n,D) ,
where V is an n-dimensional vector space over the
division ring D . Assume that G is generated by 'nice'
transformations. The problem is then to try to determine
(up to GL(V) -conjugacy) all possibilities for G . Of course,
this problem is very vague. But it is a classical one,
going back 150 years, and yet very much alive today."
— William M. Kantor, "Generation of Linear Groups,"
pp. 497-509 in The Geometric Vein: The Coxeter Festschrift ,
published by Springer, 1981
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